Local-field theory for optical phase conjugation by degenerate four wave mixing in mesoscopic interaction volumes of condensed media

A local-held theory describing optical phase conjugation in condensed media in the special case of degenerate four wave mixing is established The aim of the theory is to form the framework for microscopic studies of optical phase conjugation (i) of evanescent fields in the context of near-held optics, (ii) in mesoscopic films and quantum wells, (iii) in small particles, and (iv) in lossy media where the field penetration depth is comparable to or (substantially) less than the vacuum wavelength of the driving field. The aforementioned goal makes it necessary to abandon both the slowly varying envelope- and the electric dipole approximations usually adopted in phase conjugation studies where spatially slowly decaying or modulated held are mixed By keeping in the interaction Hamiltonian the term of second order in the vector potential and in the current-density operator the term of fist order in the vector potential new microscopic field-matter interaction processes of particular importance in the present context are included The physics of the various nonlinear microscopic processes is analysed, and systematised by presenting in diagrammatic form the nonlocal electrodynamics hidden in the nonlinear constitutive relation. A new nonlocal conductivity tensor, enabling one to describe the degenerate four wave mixing process among the prevailing local fields, is presented and the eigensymmetries of its various parts are analysed. Starting from the general local-field theory a degenerate four wave mixing response tensor of relevance for media exhibiting two-dimensional translational invariance is established and discussed. In the last part of the paper an integral equation allowing one to obtain the phase conjugated local field inside and outside the nonlinear medium is established, discussed and formally solved, and it is pointed out that this equation for systems with two-dimensional translational invariance often can be analysed analytically and numerically using methods previously developed in theoretical studies of the linear local-held electrodynamics of mesoscopic films.